求解几类复杂优化问题的进化算法及其应用
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摘要
在工程设计、科学计算、经济领域中存在着各种类型的复杂优化问题,通常需要求解它们的全局最优解而不是局部最优解。进化算法是一类鲁棒性强的全局搜索算法,对基于梯度的传统优化方法无法或难以处理的高度非线性、不可微、多峰、多变量问题,尤其是目标函数的导数无法求出,受噪声影响或没有明确的数学形式这样的问题,进化算法具有很大的优势。本文对几类复杂优化问题,包括无约束全局优化、约束优化、整数规划和混合整数规划、线性两层规划、凸二次两层规划、一般的非线性两层规划以及离散两层规划进行了系统深入的研究,根据不同类型的问题,设计和开发了不同的进化算法,并将所设计的部分算法应用到实际问题中,以验证算法的有效性和实用性。论文的主要研究成果可概括为以下几个方面:
1.在设计新变异算子的基础上提出了一种实数编码遗传算法,结合一种局部搜索算子——简化的二次插值法,提出了求解无约束全局优化问题的混合遗传算法。还分析了该算法的收敛性,对23个标准测试函数作了数值实验。与文献中其它进化算法的比较结果表明了算法的有效性。最后,将该算法应用于圆柱共形天线阵方向图优化综合,获得了良好的效果。
2.采用精确罚函数法,把约束优化问题转化为无约束优化问题,将简化的二次插值法作为局部搜索算子,合并到差分进化算法中,提出了一种求解约束优化问题的混合差分进化算法。采用该算法求解了13个标准测试问题,并与文献中已有的进化算法作了比较,结果表明混合差分进化算法性能优良。最后,采用该算法优化设计了4个著名的工程问题,所得结果表明了算法的有效性和实用性。
3.对不同类型的离散优化问题提出了不同的进化算法。首先,对一类0-1整数规划——多维背包问题,将正交试验设计应用到杂交算子中,结合一种基于贪婪思想的约束处理,提出了一种0-1编码的正交遗传算法。其次,对一般的整数规划问题,设计了一种整数编码的混合进化算法。该算法中采用带有随机参数的差分变异算子;将正交表和因素分析引入杂交过程,设计了杂交算子;设计了一种迁移算子来增加种群的多样性;把简化的二次插值法用作局部搜索算子;通过对每一代中最好染色体的基因进行微调,来跳出局部最优解。对所有算子产生的染色体均使用了一种取整和截断操作过程,以保证所求问题的变量满足整数要求和边界约束条件。最后,对混合整数规划,设计了一种整数和实数混合编码的混合进化算法。对所提出的三个进化算法作了大量的数值实验,并与文献中的其它进化算法作了比较,结果表明了三个算法的有效性。采用正交遗传算法求解了投资决策和成本预算问题,进一步说明了算法的有效性和实用性。
4.对线性两层规划和凸二次两层规划问题分别提出了一种正交遗传算法。采用KKT最优性条件将这两类两层问题分别转化为单层互补松弛问题,并对KKT乘子进行0-1二进制编码,将互补松弛问题分别转化为一系列线性规划或凸二次规划,其最优解就对应原两层规划的一个可行解。在此基础上,分别设计了求解两类单层互补松弛问题的正交遗传算法。理论分析和数值实验结果表明,求解这两类两层问题的正交遗传算法是有效的,能以较少的计算代价求出问题的全局最优解。最后,采用线性两层规划的正交遗传算法求解了价格控制问题。
5.对上层规划是非凸、不可微的非线性两层规划问题,采用均匀设计法产生初始种群,结合局部搜索算子——简化的二次插值法,提出了一种混合遗传算法。分析了算法的收敛性,大量的数值实验结果验证了算法的有效性,也表明了算法能找到高质量的最优解。最后,采用该算法求解了交通网络设计和公路通行费设置问题,进一步验证了算法的有效性和实用性。
6.对离散两层规划问题提出了两种进化算法。对整数线性两层规划问题,首先将其转化为等价的0-1线性两层规划问题,对给定的每一个上层变量,采用分支定界法求解下层问题的最优解,然后将其转化为单层隐规划问题,设计了一种正交遗传算法来求解。对上层变量是整数变量,下层变量是连续变量的混合整数非线性两层规划问题,对给定的每一个上层变量,采用传统优化算法求解下层问题的最优解,可以将两层规划转化为整数变量的单层隐规划问题,设计了一种混合进化算法来求解。数值实验结果验证了两种算法的有效性。
Complex optimization problems arise in almost every field such as engineering, science and business. A global optimal solution to these problems rather than a local optimal solution is desired. Evolutionary algorithms have been shown to be one kind of global and robust methods for solving highly nonlinear, nondifferentiable, multimodal and multivariate optimization problems. Furthermore, they are even suitable for problems whose derivatives are not available, affected by noise or whose objective functions cannot be expressed in explicit mathematical forms. These problems cannot be solved by traditional gradient-based optimization techniques. This dissertation is focused on evolutionary algorithms for several complex optimization problems, including unconstrained global optimization, constrained optimization, integer or mixed-integer programming, linear bilevel programming, convex quadratic bilevel programming, general nonlinear bilevel programming and discrete bilevel programming, and their applications. The main contributions on these subjects can be summarized as follows.
1. For solving unconstrained global optimization, a hybrid genetic algorithm is proposed. First, based on a new mutation operator, a real-coded genetic algorithm is designed. The simplified quadratic interpolation (SQI) method is then integrated into the genetic algorithm to improve its local search ability and the accuracy of the minimum function value. The theoretic analysis and simulation results on 23 benchmark functions show that the hybrid genetic algorithm is more effective and find much better solutions with lower computational cost, compared to other existing algorithms. At last, the far-field radiation patterns of the conformal antenna array are synthesized using the hybrid genetic algorithm with satisfactory results.
2. For solving constrained optimization, a hybrid differential evolution algorithm is proposed. By associating the exact penalty with all constraint violations, a constrained problem is transformed into an unconstrained problem. The SQI method is used as a local search operator for enhancing the performance of the standard differential evolution (DE) algorithm. 13 well-known benchmark problems are used to validate its efficiency. Simulation results show that the hybrid DE is superior to the standard DE. The results obtained are very competitive when comparing the proposed algorithm against other existing algorithms. At last, the algorithm is applied to four engineering design problems to demonstrate its effectiveness and applicability.
3. For three types of discrete optimization, different evolutionary algorithms are proposed. Firstly, an orthogonal genetic algorithm based on the orthogonal experimental design is proposed for solving the multidimensional knapsack problems, a kind of 0-1 integer programming problems. A check-and-repair operator based on the greedy algorithm is used to handle constraints. Secondly, an integer-coded hybrid evolutionary algorithm is developed to solve general integer programming. In this algorithm, a differential mutation operator with random parameters is adopted. A crossover operator with the orthogonal array and factor analysis is used to generate the better offspring. A migration operator is employed to keep the population’s diversity. The SQI method is also taken as a local search operator. Moreover, a few of foreign chromosomes introduced into next population are generated by randomly perturbing the best chromosome in the current population. A rounding and truncation procedure is incorporated in the operations of the algorithm to ensure that the integer restrictions and box constraints imposed on the variables are satisfied. Finally, a mixed-coding scheme of hybrid evolutionary algorithm based on the orthogonal experimental design is developed to deal with the mixed-integer programming problems. Many numerical examples are used to validate their effectiveness. Two practical examples, the investment problem and the budget problem, are provided to demonstrate the effectiveness and applicability of the orthogonal genetic algorithm.
4. For linear bilevel programming and convex quadratic bilevel programming problems (BLPPs), the orthogonal genetic algorithms are proposed. By using Karush-Kuhn-Tucker (KKT) conditions, these two BLPPs are replaced by two single-level complementary slackness problems respectively. In order to cope with the complementarity constraints, a binary encoding technology is adopted for KKT multipliers. Thus, two orthogonal genetic algorithms are presented to optimize the binary strings. Some numerical examples from the literature are used to test these methods. The theoretic analysis and experimental results show that the proposed methods are effective and can find global optimal solutions with less computation burden. At last, the price control problem is solved by the orthogonal genetic algorithm for the linear BLPP to demonstrate its effectiveness.
5. For general nonlinear BLPPs with nonconvex and nondifferentiable upper level objective function, a hybrid genetic algorithm is presented. In this algorithm, the uniform design is used to generate initial population. The SQI method is taken as a local search operator to improve solution accuracy and computational efficiency. The theoretic analysis and the simulation results on many numerical examples show that the proposed algorithm is effective and can find high quality solutions. At last, the hybrid genetic algorithm is applied to the network design problem and the toll-setting problem.
6. Two evolutionary algorithms for the discrete BLPP are presented. The discrete linear BLPP is firstly transformed into a 0-1 BLPP in which the lower level problem can be solved by the branch and bound algorithm, and then the problem is transformed into a single level 0-1 programming, which is solved by the orthogonal genetic algorithm. In addition, for the discrete nonlinear BLPP with discrete upper level variables and continuous lower level variables, the lower level problem can be solved by the traditional optimization algorithms. This bilevel problem is then transformed into a single level discrete programming problem, which is solved by the hybrid evolutionary algorithm. Some numerical examples are used to test their performance. The simulation results show that two evolutionary algorithms are effective.
1.在设计新变异算子的基础上提出了一种实数编码遗传算法,结合一种局部搜索算子——简化的二次插值法,提出了求解无约束全局优化问题的混合遗传算法。还分析了该算法的收敛性,对23个标准测试函数作了数值实验。与文献中其它进化算法的比较结果表明了算法的有效性。最后,将该算法应用于圆柱共形天线阵方向图优化综合,获得了良好的效果。
2.采用精确罚函数法,把约束优化问题转化为无约束优化问题,将简化的二次插值法作为局部搜索算子,合并到差分进化算法中,提出了一种求解约束优化问题的混合差分进化算法。采用该算法求解了13个标准测试问题,并与文献中已有的进化算法作了比较,结果表明混合差分进化算法性能优良。最后,采用该算法优化设计了4个著名的工程问题,所得结果表明了算法的有效性和实用性。
3.对不同类型的离散优化问题提出了不同的进化算法。首先,对一类0-1整数规划——多维背包问题,将正交试验设计应用到杂交算子中,结合一种基于贪婪思想的约束处理,提出了一种0-1编码的正交遗传算法。其次,对一般的整数规划问题,设计了一种整数编码的混合进化算法。该算法中采用带有随机参数的差分变异算子;将正交表和因素分析引入杂交过程,设计了杂交算子;设计了一种迁移算子来增加种群的多样性;把简化的二次插值法用作局部搜索算子;通过对每一代中最好染色体的基因进行微调,来跳出局部最优解。对所有算子产生的染色体均使用了一种取整和截断操作过程,以保证所求问题的变量满足整数要求和边界约束条件。最后,对混合整数规划,设计了一种整数和实数混合编码的混合进化算法。对所提出的三个进化算法作了大量的数值实验,并与文献中的其它进化算法作了比较,结果表明了三个算法的有效性。采用正交遗传算法求解了投资决策和成本预算问题,进一步说明了算法的有效性和实用性。
4.对线性两层规划和凸二次两层规划问题分别提出了一种正交遗传算法。采用KKT最优性条件将这两类两层问题分别转化为单层互补松弛问题,并对KKT乘子进行0-1二进制编码,将互补松弛问题分别转化为一系列线性规划或凸二次规划,其最优解就对应原两层规划的一个可行解。在此基础上,分别设计了求解两类单层互补松弛问题的正交遗传算法。理论分析和数值实验结果表明,求解这两类两层问题的正交遗传算法是有效的,能以较少的计算代价求出问题的全局最优解。最后,采用线性两层规划的正交遗传算法求解了价格控制问题。
5.对上层规划是非凸、不可微的非线性两层规划问题,采用均匀设计法产生初始种群,结合局部搜索算子——简化的二次插值法,提出了一种混合遗传算法。分析了算法的收敛性,大量的数值实验结果验证了算法的有效性,也表明了算法能找到高质量的最优解。最后,采用该算法求解了交通网络设计和公路通行费设置问题,进一步验证了算法的有效性和实用性。
6.对离散两层规划问题提出了两种进化算法。对整数线性两层规划问题,首先将其转化为等价的0-1线性两层规划问题,对给定的每一个上层变量,采用分支定界法求解下层问题的最优解,然后将其转化为单层隐规划问题,设计了一种正交遗传算法来求解。对上层变量是整数变量,下层变量是连续变量的混合整数非线性两层规划问题,对给定的每一个上层变量,采用传统优化算法求解下层问题的最优解,可以将两层规划转化为整数变量的单层隐规划问题,设计了一种混合进化算法来求解。数值实验结果验证了两种算法的有效性。
Complex optimization problems arise in almost every field such as engineering, science and business. A global optimal solution to these problems rather than a local optimal solution is desired. Evolutionary algorithms have been shown to be one kind of global and robust methods for solving highly nonlinear, nondifferentiable, multimodal and multivariate optimization problems. Furthermore, they are even suitable for problems whose derivatives are not available, affected by noise or whose objective functions cannot be expressed in explicit mathematical forms. These problems cannot be solved by traditional gradient-based optimization techniques. This dissertation is focused on evolutionary algorithms for several complex optimization problems, including unconstrained global optimization, constrained optimization, integer or mixed-integer programming, linear bilevel programming, convex quadratic bilevel programming, general nonlinear bilevel programming and discrete bilevel programming, and their applications. The main contributions on these subjects can be summarized as follows.
1. For solving unconstrained global optimization, a hybrid genetic algorithm is proposed. First, based on a new mutation operator, a real-coded genetic algorithm is designed. The simplified quadratic interpolation (SQI) method is then integrated into the genetic algorithm to improve its local search ability and the accuracy of the minimum function value. The theoretic analysis and simulation results on 23 benchmark functions show that the hybrid genetic algorithm is more effective and find much better solutions with lower computational cost, compared to other existing algorithms. At last, the far-field radiation patterns of the conformal antenna array are synthesized using the hybrid genetic algorithm with satisfactory results.
2. For solving constrained optimization, a hybrid differential evolution algorithm is proposed. By associating the exact penalty with all constraint violations, a constrained problem is transformed into an unconstrained problem. The SQI method is used as a local search operator for enhancing the performance of the standard differential evolution (DE) algorithm. 13 well-known benchmark problems are used to validate its efficiency. Simulation results show that the hybrid DE is superior to the standard DE. The results obtained are very competitive when comparing the proposed algorithm against other existing algorithms. At last, the algorithm is applied to four engineering design problems to demonstrate its effectiveness and applicability.
3. For three types of discrete optimization, different evolutionary algorithms are proposed. Firstly, an orthogonal genetic algorithm based on the orthogonal experimental design is proposed for solving the multidimensional knapsack problems, a kind of 0-1 integer programming problems. A check-and-repair operator based on the greedy algorithm is used to handle constraints. Secondly, an integer-coded hybrid evolutionary algorithm is developed to solve general integer programming. In this algorithm, a differential mutation operator with random parameters is adopted. A crossover operator with the orthogonal array and factor analysis is used to generate the better offspring. A migration operator is employed to keep the population’s diversity. The SQI method is also taken as a local search operator. Moreover, a few of foreign chromosomes introduced into next population are generated by randomly perturbing the best chromosome in the current population. A rounding and truncation procedure is incorporated in the operations of the algorithm to ensure that the integer restrictions and box constraints imposed on the variables are satisfied. Finally, a mixed-coding scheme of hybrid evolutionary algorithm based on the orthogonal experimental design is developed to deal with the mixed-integer programming problems. Many numerical examples are used to validate their effectiveness. Two practical examples, the investment problem and the budget problem, are provided to demonstrate the effectiveness and applicability of the orthogonal genetic algorithm.
4. For linear bilevel programming and convex quadratic bilevel programming problems (BLPPs), the orthogonal genetic algorithms are proposed. By using Karush-Kuhn-Tucker (KKT) conditions, these two BLPPs are replaced by two single-level complementary slackness problems respectively. In order to cope with the complementarity constraints, a binary encoding technology is adopted for KKT multipliers. Thus, two orthogonal genetic algorithms are presented to optimize the binary strings. Some numerical examples from the literature are used to test these methods. The theoretic analysis and experimental results show that the proposed methods are effective and can find global optimal solutions with less computation burden. At last, the price control problem is solved by the orthogonal genetic algorithm for the linear BLPP to demonstrate its effectiveness.
5. For general nonlinear BLPPs with nonconvex and nondifferentiable upper level objective function, a hybrid genetic algorithm is presented. In this algorithm, the uniform design is used to generate initial population. The SQI method is taken as a local search operator to improve solution accuracy and computational efficiency. The theoretic analysis and the simulation results on many numerical examples show that the proposed algorithm is effective and can find high quality solutions. At last, the hybrid genetic algorithm is applied to the network design problem and the toll-setting problem.
6. Two evolutionary algorithms for the discrete BLPP are presented. The discrete linear BLPP is firstly transformed into a 0-1 BLPP in which the lower level problem can be solved by the branch and bound algorithm, and then the problem is transformed into a single level 0-1 programming, which is solved by the orthogonal genetic algorithm. In addition, for the discrete nonlinear BLPP with discrete upper level variables and continuous lower level variables, the lower level problem can be solved by the traditional optimization algorithms. This bilevel problem is then transformed into a single level discrete programming problem, which is solved by the hybrid evolutionary algorithm. Some numerical examples are used to test their performance. The simulation results show that two evolutionary algorithms are effective.
引文
[1] D. E. Goldberg. Genetic algorithms in search, optimization and machine learning [M]. Reading, Ma: Addison Wesley, 1989.
[2] L. Davis. Handbook of Genetic Algorithms [M]. New York: Van Nostrand Reinhold, 1991.
[3] T. B?ck. Evolutionary Algorithms in Theory and Practice [M]. New York: Oxford University Press, 1996.
[4]玄光男,程润伟.遗传算法与工程优化[M].北京:清华大学出版社, 2003.
[5]李敏强,寇纪淞,林丹,李书全.遗传算法的基本理论与应用[M].北京:科学出版社, 2002.
[6]王凌.智能优化算法及其应用[M].北京:清华大学出版社, 2003.
[7]陈宝林.最优化理论与算法[M].北京:清华大学出版社, 1998.
[8]陈开周.最优化计算方法[M].西安:西安电子科技大学出版社, 1990.
[9]袁亚湘,孙文瑜.最优化理论与方法[M].北京:科学出版社, 2003.
[10] R. Horst, P. M. Pardalos, N. V. Thoai.全局优化引论[M].北京:清华大学出版社, 2003.
[11] K. Olav. Foundations of modern probability [M]. New York: Springer-Verlag , 1997.
[12] B. Patrick. Convergence of probability measures [M]. New York: Wiley , 1999.
[13] O. Hrstka, A. Ku?erová, M. Lep?, et al. A competitive comparison of different types of evolutionary algorithms [J]. Computers and Structures, 2003, 81: 1979-1990.
[14] Xin Yao, Yong Liu, and Guangming Lin. Evolutionary programming made faster [J]. IEEE Trans. Evol. Comput., 1999, 3(2): 82-102.
[15] C. Y. Lee and X. Yao. Evolutionary programming using mutations based on the Lévy probability distribution [J]. IEEE Trans. Evol. Comput., 2004, 8(1): 1-13.
[16] Ond?ej Hrstka, Anna Ku?erová. Improvements of real coded genetic algorithms based on differential operators preventing premature convergence [J]. Advances in Engineering Software, 2004, 35: 237-246.
[17] Ph. Preux, E.-G. Talbi. Towards hybrid evolutionary algorithms [J]. International transactions in operational research, 1999, 6: 557-570.
[18] Hsiao-Fan Wang and Kuang-yao Wu. Hybrid genetic algorithm for optimization problems with permutation property [J]. Computers & Operations Research, 2004, 31: 2453–2471.
[19] Lingyun Wei, Mei Zhao. A niche hybrid genetic algorithm for global optimization of continuous multimodal functions [J]. Applied Mathematics and Computation, 2005, 160(3): 649-661.
[20] R. G. Regis and C. A. Shoemaker. Local function approximation in evolutionary algorithms for the optimization of costly functions [J]. IEEE Trans. Evol. Comput., 2004, 8(5): 490-505.
[21] Yaochu Jin and Bernhard Sendhoff. Reducing Fitness Evaluations Using Clustering Techniques and Neural Network Ensembles [C]. GECCO 2004, LNCS 3102, 2004, 688–699.
[22] Alain Ratle. Accelerating the convergence of evolutionary algorithms by fitness landscape approximation [C]. PPSN V, LNCS 1498, Springer-Verlag Berlin Heidelberg 1998, 87-96.
[23] J. Branke, C. Schmidt. Faster convergence by means of fitness estimation [J]. Soft Computing, 2005, 9: 13-20.
[24] Zhenguo Tu and Yong Lu. A robust stochastic genetic algorithm (StGA) for global numerical optimization [J]. IEEE Trans. Evol. Comput., 2004, 8(5): 456-470.
[25] Lothar M. Schmitt. Theory of genetic algorithms II: models for genetic operators over the string-tensor representation of populations and convergence to global optima for arbitrary fitness function under scaling [J]. Theoretical Computer Science, 2004, 310: 181-231.
[26]徐宗本,陈志平,章祥荪.遗传算法基础理论研究的新近发展[J].数学进展, 2000, 29 (2): 97-114.
[27]郭崇慧,唐焕文.演化策略的全局收敛性[J].计算数学, 2001, 23 (1): 105-110.
[28]王登刚,刘迎曦,李守巨.最优化问题全局寻优的混合遗传算法[J].力学学报, 2002, 34 (3): 469-474.
[29]郭观七,喻寿益,贺素良.自适应小生态遗传算法的理论分析和加速技术[J].计算机学报, 2003, 26(6): 753-758.
[30]王宇平,刘大莲.基于平滑技术和一维搜索的全局优化进化算法及其收敛性[J].计算机学报, 2006, 29(4): 670-675.
[31]王凌,何锲,金以慧.智能约束处理技术综述[J].化工自动化及仪表, 2008, 35 (1): 1-7.
[32] M. M. Ali, A. T?rn, S. Viitanen. A numerical comparison of some modified controlled random search algorithms [J]. Journal of Global Optimization, 1997, 11: 377-385.
[33] P. Brachetti, M. De Felice Ciccoli, G. Di Pillo and S. Lucidi. A new version of the Price’s algorithm for global optimization [J]. Journal of Global Optimization, 1997, 10: 165-184.
[34] Jiao Yong-Chang, Dang Chuangyin, Leung Yee and Hao Yue. A modification to the new version of the Price's algorithm for continuous global optimization problems [J]. Journal of Global Optimization, 2006, 36(4): 609-626.
[35] Q. Zhang, J. Sun, E. Tsang and J. Ford. Hybrid estimation of distribution algorithm for global optimization [J]. Engineering Computations, 2004, 21(1): 91-107.
[36] D. Goldberg and S. Voessner. Optimizing global-local search hybrids [C]. In Proc. Genetic Evol. Comput. Conf. (GECCO’99), 1999, 220-228.
[37] C. A. C. Coello. Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: A survey of the state of the art [J]. Comput.Methods Appl. Mech. Eng., 2002, 191(11-12): 1245-1287.
[38] J. H. Kim, and H. Myung. Evolutionary programming techniques for constrained optimization problems [J]. IEEE Trans. on Evol. Comput., 1997, 11(2): 129-140.
[39] T. P. Runarsson, and X. Yao. Stochastic ranking for constrained evolutionary optimization [J]. IEEE Trans. on Evolutionary Computation, 2000, 4(3): 284-294.
[40] T. P. Runarsson, and X. Yao. Search biases in constrained evolutionary optimization [J]. IEEE Trans. Syst., Man, Cybern. C, Appl. Rev.,2005, 35(2): 233-243.
[41] E. Mezura-Montes, and C.A.C. Coello. A simple multimembered evolution strategy to solve constrained optimization problems [J]. IEEE Trans. on Evol. Comput., 2005, 9(1): 1-17.
[42] S. Venkatraman and G. G. Yen. A generic framework for constrained optimization using genetic algorithms [J]. IEEE Trans. on Evol. Comput., 2005, 9(4): 424-435.
[43] T. Takahama, and S. Sakai. Constrained optimization by applying theαconstrained method to the nonlinear simplex method with mutations [J]. IEEE Trans. on Evol. Comput., 2005, 9(5): 437-451.
[44] Z. Cai and Y. Wang. A multiobjective optimization-based evolutionary algorithm for constrained optimization [J]. IEEE Trans. on Evol. Comput., 2006, 10(6): 658-675,.
[45] Y. Wang, Z. Cai, G. Guo, and Y. Zhou. Multiobjective optimization and hybrid evolutionary algorithm to solve constrained optimization problems [J]. IEEE Trans. Syst., Man, Cybern. B, Cybern, 2007, 37(3): 560-575.
[46] Y. Wang, Z. Cai, Y. Zhou, et al.. An adaptive tradeoff model for constrained evolutionary optimization [J]. IEEE Trans. On Evol. Comput., 2008, 12(1): 80-92.
[47] Helio J. C. Barbosa, Afonso C. C. Lemonge. A new adaptive penalty scheme for genetic algorithms [J]. Information Sciences, 2003, 156: 215-251.
[48] R. Storn and K. Price. Differential evolution-A simple and efficient heuristic for global optimization over continuous spaces [J]. Journal of Global Optimization, 1997, 11: 341-359.
[49] J. Sun, Q. Zhang, and E. Tsang. DE/EDA: A new evolutionary algorithm for global optimization [J]. Information Sciences, 2005, 169(3): 249-262.
[50] J. Brest, S. Greiner, et al.. Self-adapting control parameters in differential evolution: A comparative study on numerical benchmark problems [J]. IEEE Trans. on Evol. Comput., 2006, 10(6): 646-657.
[51] N. Noman, H. Iba. Accelerating differential evolution using an adaptive local search [J], IEEE Trans. on Evol. Comput., 2008, 12(1): 107-125.
[52] S. Rahnamayan, H. R. Tizhoosh, and M. M. A. Salama. Opposition-based differential evolution [J]. IEEE Trans. on Evol. Comput., 2008, 12(1): 64-79.
[53] Chang S, H-g C. Optimal approximation of linear systems by a differential evolution algorithm [J]. IEEE Trans. on Syst., Man, Cybern. A, 2001, 31(6): 698-707.
[54] U. K. Chakraborty, S. Das, A. Konar. Differential evolution with local neighborhood [C]. 2006 IEEE Congress on Evolutionary Computation, 2006, 2042-2049.
[55] T. Takahama, S. Sakai. Constrained optimization by theεconstrained differential evolution with gradient-based mutation and feasible elites [C]. 2006 IEEE Congress on Evolutionary Computation, 2006, 1-8.
[56] V. L. Huang, A. K. Qin, P. N. Suganthan. Self-adaptive differential evolution algorithm for constrained real-parameter optimization [C]. 2006 IEEE Congress on Evolutionary Computation, 2006, 17-24.
[57] E. Mezura-Montes, J. Velázquez-Reyes, C. A. C. Coello. Modified differential evolution for constrained optimization [C]. 2006 IEEE Congress on Evolutionary Computation, 2006, 25-32.
[58] J. J. Liang, T. P. Runarsson, E. Mezura-Montes, et al. Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization [R]. Technical Report, 2005. http://www.ntu.edu.sg/home/EPNSugan.
[59] E. M. Montes, C. A. C. Coello, and R. L. Becerra. Engineering optimization using a simple evolutionary algorithm [C]. In Proc. 15th Int. Conf. Tools with Artif. Intell., 2003, 149-156.
[60] H. K. Kim, J. K. Chong, K. Y. Park, and D. A. Lowther. Differential evolution strategy for constrained global optimization and application to practical engineering problems [J]. IEEE Trans. on Magnetics, 2007, 43(4): 1565-1568.
[61] Qingfu Zhang and Yiu-Wing Leung. An orthogonal genetic algorithm for multimedia multicast routing [J]. IEEE Trans. on Evol. Comput., 1999, 3(1): 53-62.
[62] Yiu-wing Leung, and Yuping Wang. An orthogonal genetic algorithm with quantization for global numerical optimization [J]. IEEE Trans. on Evol. Comput., 2001, 5(1): 53-62.
[63] Jinn-Tsong Tsai, Tung-Kuan Liu, and Jyh-Horng Chou. Hybrid Taguchi-genetic algorithm for global numerical optimization [J]. IEEE Trans. on Evol. Comput. 2004, 8(4): 365-377.
[64] Shinn-Ying Ho, Li-Sun Shu, and Jian-Hung Chen. Intelligent Evolutionary Algorithms for Large Parameter Optimization Problems [J]. IEEE Trans. on Evol. Comput., 2004, 8(6): 522-540.
[65] Carlos Cotta, Jose Ma Troya. A Hybrid Genetic Algorithm for the 0-1 Multiple Knapsack Problem [C]. Artificial Neural Nets and Genetic Algorithms 3, Springer-Verlag Wien New York 1998, 250-254.
[66] Jian-cong Bai, Hui-you Chang, Yang Yi. A Partheno-Genetic Algorithm for Multidimensional Knapsack Problem [C]. Proceedings of the Fourth International Conference on Machine Learning and Cybernetics, 2005, 2962-2965.
[67] Ng Chi-Kong, Zhang Lian-Sheng, Li Duan, and Tian Wei-Wen. Discrete filled function method for discrete global optimization [J]. Computational Optimization and Applications, 2005, 31: 87-115.
[68] C. Mohan, H. T. Nguyen. A controlled random search technique incorporating the simulated annealing concept for solving integer and mixed integer global optimization Problems [J]. Computational Optimization and Applications, 1999, 14: 103-132.
[69] K. E. Parsopoulos, and M. N. Vrahatis. Recent approaches to global optimization problems through particle swarm optimization [J]. Natural Computing, 2002, 1: 235-306.
[70] Lino Costa, Pedro Oliveira. Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems [J]. Computers and Chemical Engineering, 2001, 25: 257–266.
[71] Y. C. Lin, and K. S. Hwang. A mixed-coding scheme of evolutionary algorithms to solve mixed-integer nonlinear programming problems [J]. Computers and Mathematics with Applications, 2004, 47: 1295-1307.
[72] I. E. Grossmann. Review of nonlinear mixed-integer and disjunctive programming techniques [J]. Optimization and Engineering, 2002, 3: 227-252.
[73] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu. Finding multiple solutions to general integer linear programs [J]. European Journal of Operational Research, 2008, 184: 802-809.
[74] L. T. Biegler, I. E. Grossmann. Retrospective on optimization [J]. Computers and Chemical Engineering, 2004, 28: 1169–1192.
[75] I. E. Grossmann, L. T. Biegler. Future perspective on optimization [J]. Computers and Chemical Engineering, 2004, 28: 1193–1218.
[76]朱文兴.整数规划的一类填充函数算法[J].应用数学学报, 2000, 23 (4): 481-487.
[77]方开泰,马长兴.正交与均匀试验设计[M].北京:科学出版社, 2001.
[78]邢文训,谢金星.现代优化计算方法[M].北京:清华大学出版社, 2000.
[79]陈志平等.求解中大规模复杂凸二次整数规划问题的新型分枝定界算法[J].计算数学, 2004, 26 (4): 445-458.
[80]杜海峰,刘若辰,焦李成等.求解0-1背包问题的人工免疫抗体修正克隆算法[J].控制理论与应用, 2005, 22 (3): 348-352.
[81]王宇平,焦永昌,张福顺.解多目标优化的均匀正交遗传算法[J].系统工程学报, 2003, 18 (6): 481-486.
[82] Rainer Storn. System design by constraint adaptation and differential evolution [J]. IEEE Trans. on Evolutionary Computation, 1999, 3 (1): 22-34.
[83] Karin Zielinski, and Rainer Laur. Constrained single-objective optimization using differential evolution [C]. 2006 IEEE Congress on Evolutionary Computation, CEC2006, Vancouver, BC, Canada, 2006, pp. 223-230.
[84] M. F. Tasgetiren, and P. N. Suganthan. A multi-populated differential evolution algorithm for solving constrained optimization problem [C]. 2006 IEEE Congress on Evolutionary Computation, CEC2006, Vancouver, BC, Canada, 2006, pp. 33-40.
[85] Janez Brest, Viljem Zumer, and Mirjam Sepesy Mau?ec. Self-adaptive differential evolution algorithm in constrained real-parameter optimization [C].2006 IEEE Congress on Evolutionary Computation, CEC2006, Vancouver, BC, Canada, 2006, pp. 215-222.
[86] G. Rudolph. Convergence analysis of canonical genetic algorithms [J]. IEEE Trans. On Neural Networks, 1994, 5 (1): 96-101.
[87] N. Srinivas, and K. Deb. Multiobjective function optimization using nondominated sorting genetic algorithms [J]. Evolutionary Computation, 1995, 2(3): 221-248.
[88] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan. A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ[J]. IEEE Transactions on Evolutionary Computation, 2002, 6(2): 182-197.
[89] E. Zitzler, L. Thiele. Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach [J]. IEEE Trans. On Evolutionary Computa- tion, 1999, 3(4): 257-271.
[90] J. D. Knowles and D. W. Corne. Approximating the nondominated front using the pareto archived evolution strategy [J]. Evolutionary Computation, 2000, 8 (2): 149-172.
[91] Carlos A. Coello Coello. Evolutionary multi-objective optimization : a historical view of the field [J]. IEEE Computational Intelligence Magazine, 2006, 1 (1): 28-36.
[92] J. F. Bard. Practical bilevel optimization: algorithms and application [M]. The Netherlands: Kluwer Academic Publishers, 1998.
[93] S. R. Hejazi, A. Memariani, G. Jahanshanloo, M. M. Sepehri. Linear bilevel programming solution by genetic algorithm [J]. Computers & Operations Research, 2002, 29: 1913-1925.
[94] G. Anandalingam and D. J. White. A solution for the linear static Stackelberg problem using penalty function [J]. IEEE Transactions Automatic Control, 1990, 35: 1170-1173.
[95] P. Hansen, B. Jaumard and G. Savard. New branch-and-bound rules for linear bilevel programming [J]. SIAM Journal on Science and Statistical Computing, 1992, 13(5): 1194-1217.
[96] Y. H. Liu, and T. H. Spencer. Solving a bilevel linear program when the inner decision maker controls few variables [J]. European Journal of Operational Research, 1995, 81: 644-651.
[97] K. Moshirvaziri, M. A. Amouzegar and S. E. Jacobsen. Test problem construction for linear bilevel programming problems [J]. Journal of Global Optimization 1996, 8: 235-243.
[98] H. Tuy, A. Migdalas, N. T. Hoai-Phuong. A novel approach to bilevel nonlinear programming [J]. Journal of Global Optimization, 2007, 38: 527-554.
[99] Q. Wang, F. Yang, S. Wang and Y. H. Liu. Bilevel programs with multiple followers [J]. Systems Science and Mathematical Sciences, 2000, 13(3): 265-276.
[100] K. M. Lan, U. P. Wen, H. S. Shih and E. S. Lee. A hybrid neural network approach to bilevel programming problems [J]. Applied Mathematics Letters, 2007, 20(8): 880-884.
[101] J. Lu, C. Shi, G. Zhang, Tharam Dillon. Model and extended Kuhn-Tucker approach for bilevel multi-follower decision making in a referential-uncooperative situation [J]. Journal of Global Optimization, 2007, 38: 597-608.
[102] Z. H. Gümüs, and C. A. Floudas. Global optimization of nonlinear bilevel programming problems [J]. Journal of Global Optimization,2001, 20: 1-31.
[103] J. Bard. Convex two-level optimization [J]. Mathematical Programming, 1988, 40: 15-27.
[104] J. Bard and J. Moore. A branch and bound algorithm for the bilevel programming problem [J]. SIAM Journal on Scientific Statistical Computing, 1990, 11: 281-292.
[105] T. Edmunds and J. Bard. Algorithms for nonlinear bilevel mathematical programming [J]. IEEE Trans. on Systems, Man, and Cybernetics, 1991, 21: 83-89.
[106] L. D. Muu, N. V. Quy. A global optimization method for solving convex quadratic bilevel programming problems [J]. Journal of Global Optimization, 2003, 26: 199-219.
[107] K. Shimizu, and E. Aiyoshi. A new computational method for Stackelberg and min-max problems by use of a penalty method [J]. IEEE Transactions on Automatic Control, 1981, 26(2): 460-466.
[108] J. V. Outrata. On the numerical solution of a class of Stackelberg problems [J]. Zeitschrift Fur Operations Research, 1990, 34: 255-277.
[109] V. Oduguwa, and R. Roy. Bi-level optimization using genetic algorithm [C]. In proc. IEEE int. Conf. Artificial Intelligence Systems, 2002, 123-128.
[110] J. E. Falk, Jiming Liu. On bilevel programming, Part I: general nonlinear cases [J]. Mathematical Programming, 1995, 70: 47-72.
[111] P.H. Calamai and L.N. Vicente. Generating quadratic bilevel programming test problems [J]. ACM Transactions on Mathematical Software, 1994, 20(1): 103-119.
[112] L. N. Vicente and P. H. Calamai, Bilevel and multilevel programming: A bibliography review [J]. Journal of Global Optimization, 1994, 5(3): 291-306.
[113] S. Dempe. Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints [J]. Optimization, 2003, 52: 333-359.
[114] B. Colson, P. Marcotte, and G. Savard. Bilevel programming: a survey [J]. A Quarterly Journal of Operations Research(4OR), 2005, 3: 87-107.
[115] B. Colson, P. Marcotte, and G. Savard. An overview of bilevel optimization [J]. Ann. Oper. Res., 2007, 153: 235-256.
[116] BaoDing Liu. Stackelberg-Nash equilibrium for multilevel programming with multiple followers using genetic algorithms [J]. Computers Math. Applic., 1998, 36 (7): 79-89.
[117] Yuping Wang, Yong-Chang Jiao, Hong Li. An evolutionary algorithm for solving nonlinear bilevel programming based on a new constraint-handing scheme [J]. IEEE Trans. on Systems, Man, and Cybernetics-Part C, 2005, 35(2): 221-232.
[118] Beno?t Colson, Patrice Marcotte, and Gilles Savard. A trust-region method for nonlinear bilevel programing: algorithm and computational experience [J]. Computational optimization and applications, 2005, 30: 211-227.
[119] B. Colson. BIPA (Bilevel programming with approximation methods): Software guide and test problems [R]. Technical Report CRT-2002-38, Centre de Recherche sur les Transports, Universitéde Montréal, Montréal, Qc, Canada, Oct. 2002. (http//:citeseer. ist. psu.edu/ Cache).
[120] E. Aiyoshi, K. Shimizu. A solution method for the static constrained Stackelberg problem via penalty method [J]. IEEE Trans. on Automatic Control, 1984, 29(12): 1111-1114.
[121] M. A. Amouzegar. A global optimization method for nonlinear bilevel programming problems [J]. IEEE Trans. on Systems, Man, and Cybernetics-Part B: Cybernetics, 1999, 29(6): 771-777.
[122] H. I. Calvete, C. Gal. The bilevel linear/linear fractional programming problem [J]. European Journal of Operational Research, 1999, 114(1): 188-197.
[123] Z. H. Gümüs, C. A. Floudas. Global optimization of mixed-integer bilevel programming problems [J]. Computational Management Science, 2005, 2: 181-212.
[124] J. T. Moore, J. F. Bard. The mixed integer linear bilevel programming problem [J]. Operations Research, 1990, 38: 911-921.
[125] U. P. Wen, and Y. H. Yang. Algorithms for solving the mixed integer two-level linear programming problem [J]. Computers & Operations Research, 1990, 17: 133-142.
[126] U. P. Wen, A. D. Huang. A simple Tabu Search method to solve the mixed-integer linear bilevel programming problem [J]. European Journal of Operational Research, 1996, 88: 563-571.
[127] L. Vicente, G. Savard and J. Judice. Discrete linear bilevel programming problem [J]. Journal of Optimization theory and Applications, 1996, 89(3): 597-614.
[128] J. F. Bard, and J. T. Moore, An algorithm for the discrete bilevel programming problem [J]. Naval Research Logistics, 1992, 39: 419-435.
[129] S. Dempe, V. Kalashnikov, R. Z. Ríos-Mercado. Discrete bilevel programming: Application to a natural gas cash-out problem [J]. European Journal of Operational Research, 2005, 166: 469-488.
[130] T. A. Edmunds, and J. F. Bard. An algorithm for the mixed-integer nonlinear bilevel programming problem [J]. Annals of Operations Research, 1992, 34: 149-162.
[131] R. H. Jan, and M. S. Chern. Non-linear integer bilevel programming [J]. European Journal of Operational Research, 1994, 72: 574-587.
[132] L. Vicente, G. Savard and J. Judice. Descent approach for quadratic bilevel programming [J]. J. Optim. Theory and Appl., 1994, 81(2): 379-399.
[133] C. Kolstad, and L. Lasdon. Derivative evaluation and computational experience with large bilevel mathematical programs [J]. Journal of Optimization theory and Applications, 1990, 65: 485-499.
[134] E. Aiyoshi, and K. Shimizu. Hierarchical decentralized systems and its newsolution by a barrier method [J]. IEEE Trans. on Systems, Man, and Cybernetics, 1981, 11: 444-449.
[135] Y. Ishizuka and E. Aiyoshi. Double penalty method for bilevel optimization problems [J]. Annals of Operations Research, 1992, 34: 73-88.
[136] K. Shimizu and Min Lu. A global optimization Method for the Stackelberg problem with convex functions via problem transformation and concave programming [J]. IEEE Trans. on Systems, Man, and Cybernetics, 1995, 25(12): 1635-1640.
[137] J. F. Bard. Some properties of the bilevel programming problem [J]. Journal of Optimization Theory and Applications, 1991, 68(2): 371-378.
[138] A. Yezza. First-order necessary optimality conditions for general bilevel programming problems [J]. Journal of Optimization Theory and Applications, 1996, 89(1): 189-219.
[139] Y. Ishizuka. Optimality conditions for quasi-differentiable programs with application to two-level optimization [J]. SIAM J. Control and Optimization, 1988, 26(6): 1388-1398.
[140] H. I. Calvete, and C. Gale. On the quasiconcave bilevel programming problem [J]. Journal of Optimization Theory and Applications, 1998, 98(3): 613-622.
[141] J.V. Outrata. Necessary optimality conditions for Stackelberg problems [J]. Journal of Optimization Theory and Applications, 1993, 76(2): 305-320.
[142] K. Shimizu, and Y. Ishizuka. Optimality conditions and algorithms for parameter design problems with two-level structure [J]. IEEE Trans. On Automatic Control, 1985, 30(10): 986-993.
[143] P. A. Clark, and A. W. Westerberg. Bilevel programming for steady-state chemical process design-I: Fundamentals and Algorithms [J]. Computers Chem. Engng., 1990, 14(1): 87-97.
[144] P. A. Clark. Bilevel programming for steady-state chemical process design-Ⅱ: Performance study for Nondegenerate problems [J]. Computers Chem. Engng., 1990, 14(1): 99-109.
[145] Z. Gao, H. Sun, L. L. Shan. A continuous equilibrium network design model and algorithm for transit systems [J]. Transportation Research Part B, 2004, 38: 235-250.
[146] Suh-Wen Chiou. Bilevel programming for the continuous transport network design problem [J]. Transportation Research Part B, 2005, 39: 361–383.
[147]向丽.递阶优化问题理论及其算法研究与进展[J].控制与决策, 2001,16(6): 854-858.
[148]杨若黎,顾基发.一类非线性两级规划问题的模拟退火求解[J].系统工程理论与实践, 1997, 17(7):52-64.
[149] S. Dempe. A bundle algorithm applied to bilevel programming problems with non-unique lower level solutions [J]. Computational Optimization and Applications, 2000, 15: 145-166.
[150]魏祥云,王先甲.二层系统决策的最优性[J].系统工程理论与实践, 1998, 6: 7-12.
[151]谈烨,仲伟俊,徐南荣.多种资源在多项目间分配的两层决策方法[J].系统工程学报, 1999, 14(3): 290-295.
[152]杜纲.非光滑多目标Stackelberg问题的最优性条件[J].系统工程学报, 1996, 11(2): 22-28.
[153] A. G. Mersha, S. Dempe. Linear bilevel programming with upper level constraints depending on the lower level solution [J]. Applied Mathematics and Computation, 2006, 180: 247-254.
[154] G. S. Liu, J. Y. Han, J. Z. Zhang. Exact penalty functions for convex bilevel programming problems [J]. Journal of Optimization Theory and Applications, 2001, 110 (3): 621-643.
[155]邓先礼,雷万明.用罚函数求解二层凸规划的方法[J].应用数学学报,2001, 24 (2): 161-167.
[156]向丽,顾培亮.一种基于遗传算法的双层非线性多目标决策方法[J].系统工程理论方法应用, 1999, 8(3): 16-21.
[157]王广民,万仲平,王先甲.二(双)层规划综述[J].数学进展, 2007, 36 (5): 513-529.
[158]盛昭翰,吕庆喆,徐南荣.一类二层决策问题的性质分析和基于Frank-Walfe方法的神经网络算法[J].自动化学报, 1996, 22 (6): 657-665.
[159]刘树安,尹新,郑秉霖等.二层线性规划问题的遗传算法求解[J].系统工程学报, 1999, 14 (3): 280-285.
[160]仲伟俊,徐南荣.两层决策的波尔兹曼机方法[J].系统工程学报, 1995, 10(1): 7-13.
[161]王春峰,李光泉,郑丕谔.非光滑两级优化问题的必要条件及其算法[J].系统工程学报, 1998, 13 (3): 92-99.
[162] C. Audet, J. Haddad, G. Savard. A note on the definition of a linear bilevel programming solution [J]. Applied Mathematics and Computation, 2006, 181: 351-355.
[163] C. Shi, J. Lu, G. Zhang, et al. An extended branch and bound algorithm for linear bilevel programming [J]. Applied Mathematics and Computation, 2006, 180: 529-537.
[164] C. Shi, J. Lu, G. Zhang. An extended Kth-best approach for linear bilevel programming [J]. Applied Mathematics and Computation, 2005, 164: 843-855.
[165]林丹,王宏,李敏强.用多目标进化算法求解二层规划双目标模型[J].系统工程理论与实践, 2006, 5: 106-110.
[166]李宏,王宇平.解非线性两层规划的一种混合遗传算法[J].西安电子科技大学学报, 2002, 29(6): 840-843.
[167] Zhi Xu, Hong Li, Qi-zhong Liu and Jian-ying Li. Pattern synthesis of conformal antenna array by the hybrid genetic algorithm [J]. Progress in Electromagnetics Research (PIER), 2008, 79: 75-90.
[168] Shamim Akhtar, Kang Tai and Tapabrata Ray. A socio-behavioural simulation model for engineering design optimization [J]. Eng. Opt., 2002, 34: 341–354.
[169] H. I. Calvete, C. Galé, P. M. Mateo. A new approach for solving linear bilevel problems using genetic algorithms [J]. European Journal of Operational Research, 2008, 188: 14-28.
[170] C. Audet, G. Savard, W. Zghal. New Branch-and-Cut Algorithm for Bilevel Linear Programming [J]. J. Optim. Theory Appl. ,2007, 134: 353-370.
[171]左黎明,汤鹏志,廖宇波等.关于销售集团投资设置销售分店问题IP模型[J].数学的实践与认识, 2005, 35 (2): 21-25.
[172]姚瑞枫,宋玉阶.多维0-1背包问题的混合遗传算法[J].武汉科技大学学报(自然科学版),2003,26(2):214-217.
[173]周水生,刘三阳,刘红英.价格控制问题及其推广形式的罚函数法[J].系统工程学报, 1999, 14(2): 156-161.
[174]阮国桢,杨丰梅,汪寿阳.线性二级价格控制问题的单纯形算法[J].系统工程理论与实践, 1996, 12: 38-43.
[175]滕春贤,姚锋敏,陈东彦.价格控制问题精确罚等价形式的研究[J].黑龙江大学自然科学学报, 2007, 24(1): 10-15.
[176] W. Bialas,M. Karwan. Two-level linear programming [J]. Management Science, 1984, 30 : 1004-1020.
[177]李煜华,张铁男,孙凯.基于价格控制的排污收费二层线性规划模型[J].统计与决策, 2008, 11: 41-43.
[178] M. Labbé, P. Marcotte, and G. Savard. A bilevel model of taxation and its applications to optimal highway pricing [J]. Management Science, 1998, 44: 1595-1607.
[179] P. Marcotte. Network design problem with congestion effects: A case of bilevel programming [J]. Mathematical Programming, 1986, 34:142–162.
[180] A. Migdalas. Bilevel programming in traffic planning: models, methods and challenge [J]. Journal of Global Optimization, 1995, 7: 381-405.
[181] Janet Clegg, Mike Smith, Yanling Xiang, et al. Bilevel programming applied to optimising urban transportation [J]. Transportation Research Part B, 2001, 35: 41-70.
[182] W. Candler, R. Townsley. A linear two-level programming problem [J]. Computers and Operations Research, 1982, 9: 59–76.
[183]刘红英,刘三阳.有关线性价格控制问题的注记[J].系统工程理论与实践, 1999, 19 (4): 141-143.
[184]刘国山,韩继业,汪寿阳.双层优化问题的信赖域算法[J].科学通报, 1998, 43 (4): 383-387.
[185] P. Marcotte, G. Savard, D. L. Zhu. A trust region algorithm for nonlinear bilevel programming [J]. Operations Research Letters, 2001, 29: 171-179.
[186]李云雁等.实验设计与数据处理[M].北京:化学工业出版社, 2005.
[187] P. C. Chu, and J. E. Beasley. A genetic algorithm for the multidimensional knapsack problem [J]. Journal of Heuristics, 1998, 4: 63–86.
[188] D. H. Wolpert, and W. G. Macready. No free lunch theorems for optimization [J]. IEEE Trans. Evol. Comput., 1997, 1(1): 67-82.
[189] K. Deb. An efficient constraint handing method for genetic algorithms [J]. Computer Methods in Applied Mechanics and Engineering, 2000, 186 (2/4): 311-338.
[190]方开泰,王元.数论方法在统计中的应用[M].北京:科学出版社, 1996.
[191]高齐圣,潘德惠.基于均匀设计的遗传算法及其应用[J].信息与控制, 1999, 28(3): 236-240.
[192] S. Dempe. A necessary and a sufficient optimality condition for bilevel programming problems [J]. Optimization, 1992, 25: 341-354.
[193] D. L. Zhu, Q. Xu, Z. Lin. A homotopy method for solving bilevel programming problem [J]. Nonlinear Analysis, 2004, 57: 917-928.
[194] S. Dempe. First-Order Necessary Optimality Conditions for General Bilevel Programming Problems [J]. Journal of Optimization Theory and Applications, 1997, 95(3): 735-739.
[195] N. P. Faísca, V. Dua, B. Rustem, et al. Parametric global optimisation for bilevel programming [J]. Journal of Global Optimization, 2007, 38: 609-623.
[2] L. Davis. Handbook of Genetic Algorithms [M]. New York: Van Nostrand Reinhold, 1991.
[3] T. B?ck. Evolutionary Algorithms in Theory and Practice [M]. New York: Oxford University Press, 1996.
[4]玄光男,程润伟.遗传算法与工程优化[M].北京:清华大学出版社, 2003.
[5]李敏强,寇纪淞,林丹,李书全.遗传算法的基本理论与应用[M].北京:科学出版社, 2002.
[6]王凌.智能优化算法及其应用[M].北京:清华大学出版社, 2003.
[7]陈宝林.最优化理论与算法[M].北京:清华大学出版社, 1998.
[8]陈开周.最优化计算方法[M].西安:西安电子科技大学出版社, 1990.
[9]袁亚湘,孙文瑜.最优化理论与方法[M].北京:科学出版社, 2003.
[10] R. Horst, P. M. Pardalos, N. V. Thoai.全局优化引论[M].北京:清华大学出版社, 2003.
[11] K. Olav. Foundations of modern probability [M]. New York: Springer-Verlag , 1997.
[12] B. Patrick. Convergence of probability measures [M]. New York: Wiley , 1999.
[13] O. Hrstka, A. Ku?erová, M. Lep?, et al. A competitive comparison of different types of evolutionary algorithms [J]. Computers and Structures, 2003, 81: 1979-1990.
[14] Xin Yao, Yong Liu, and Guangming Lin. Evolutionary programming made faster [J]. IEEE Trans. Evol. Comput., 1999, 3(2): 82-102.
[15] C. Y. Lee and X. Yao. Evolutionary programming using mutations based on the Lévy probability distribution [J]. IEEE Trans. Evol. Comput., 2004, 8(1): 1-13.
[16] Ond?ej Hrstka, Anna Ku?erová. Improvements of real coded genetic algorithms based on differential operators preventing premature convergence [J]. Advances in Engineering Software, 2004, 35: 237-246.
[17] Ph. Preux, E.-G. Talbi. Towards hybrid evolutionary algorithms [J]. International transactions in operational research, 1999, 6: 557-570.
[18] Hsiao-Fan Wang and Kuang-yao Wu. Hybrid genetic algorithm for optimization problems with permutation property [J]. Computers & Operations Research, 2004, 31: 2453–2471.
[19] Lingyun Wei, Mei Zhao. A niche hybrid genetic algorithm for global optimization of continuous multimodal functions [J]. Applied Mathematics and Computation, 2005, 160(3): 649-661.
[20] R. G. Regis and C. A. Shoemaker. Local function approximation in evolutionary algorithms for the optimization of costly functions [J]. IEEE Trans. Evol. Comput., 2004, 8(5): 490-505.
[21] Yaochu Jin and Bernhard Sendhoff. Reducing Fitness Evaluations Using Clustering Techniques and Neural Network Ensembles [C]. GECCO 2004, LNCS 3102, 2004, 688–699.
[22] Alain Ratle. Accelerating the convergence of evolutionary algorithms by fitness landscape approximation [C]. PPSN V, LNCS 1498, Springer-Verlag Berlin Heidelberg 1998, 87-96.
[23] J. Branke, C. Schmidt. Faster convergence by means of fitness estimation [J]. Soft Computing, 2005, 9: 13-20.
[24] Zhenguo Tu and Yong Lu. A robust stochastic genetic algorithm (StGA) for global numerical optimization [J]. IEEE Trans. Evol. Comput., 2004, 8(5): 456-470.
[25] Lothar M. Schmitt. Theory of genetic algorithms II: models for genetic operators over the string-tensor representation of populations and convergence to global optima for arbitrary fitness function under scaling [J]. Theoretical Computer Science, 2004, 310: 181-231.
[26]徐宗本,陈志平,章祥荪.遗传算法基础理论研究的新近发展[J].数学进展, 2000, 29 (2): 97-114.
[27]郭崇慧,唐焕文.演化策略的全局收敛性[J].计算数学, 2001, 23 (1): 105-110.
[28]王登刚,刘迎曦,李守巨.最优化问题全局寻优的混合遗传算法[J].力学学报, 2002, 34 (3): 469-474.
[29]郭观七,喻寿益,贺素良.自适应小生态遗传算法的理论分析和加速技术[J].计算机学报, 2003, 26(6): 753-758.
[30]王宇平,刘大莲.基于平滑技术和一维搜索的全局优化进化算法及其收敛性[J].计算机学报, 2006, 29(4): 670-675.
[31]王凌,何锲,金以慧.智能约束处理技术综述[J].化工自动化及仪表, 2008, 35 (1): 1-7.
[32] M. M. Ali, A. T?rn, S. Viitanen. A numerical comparison of some modified controlled random search algorithms [J]. Journal of Global Optimization, 1997, 11: 377-385.
[33] P. Brachetti, M. De Felice Ciccoli, G. Di Pillo and S. Lucidi. A new version of the Price’s algorithm for global optimization [J]. Journal of Global Optimization, 1997, 10: 165-184.
[34] Jiao Yong-Chang, Dang Chuangyin, Leung Yee and Hao Yue. A modification to the new version of the Price's algorithm for continuous global optimization problems [J]. Journal of Global Optimization, 2006, 36(4): 609-626.
[35] Q. Zhang, J. Sun, E. Tsang and J. Ford. Hybrid estimation of distribution algorithm for global optimization [J]. Engineering Computations, 2004, 21(1): 91-107.
[36] D. Goldberg and S. Voessner. Optimizing global-local search hybrids [C]. In Proc. Genetic Evol. Comput. Conf. (GECCO’99), 1999, 220-228.
[37] C. A. C. Coello. Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: A survey of the state of the art [J]. Comput.Methods Appl. Mech. Eng., 2002, 191(11-12): 1245-1287.
[38] J. H. Kim, and H. Myung. Evolutionary programming techniques for constrained optimization problems [J]. IEEE Trans. on Evol. Comput., 1997, 11(2): 129-140.
[39] T. P. Runarsson, and X. Yao. Stochastic ranking for constrained evolutionary optimization [J]. IEEE Trans. on Evolutionary Computation, 2000, 4(3): 284-294.
[40] T. P. Runarsson, and X. Yao. Search biases in constrained evolutionary optimization [J]. IEEE Trans. Syst., Man, Cybern. C, Appl. Rev.,2005, 35(2): 233-243.
[41] E. Mezura-Montes, and C.A.C. Coello. A simple multimembered evolution strategy to solve constrained optimization problems [J]. IEEE Trans. on Evol. Comput., 2005, 9(1): 1-17.
[42] S. Venkatraman and G. G. Yen. A generic framework for constrained optimization using genetic algorithms [J]. IEEE Trans. on Evol. Comput., 2005, 9(4): 424-435.
[43] T. Takahama, and S. Sakai. Constrained optimization by applying theαconstrained method to the nonlinear simplex method with mutations [J]. IEEE Trans. on Evol. Comput., 2005, 9(5): 437-451.
[44] Z. Cai and Y. Wang. A multiobjective optimization-based evolutionary algorithm for constrained optimization [J]. IEEE Trans. on Evol. Comput., 2006, 10(6): 658-675,.
[45] Y. Wang, Z. Cai, G. Guo, and Y. Zhou. Multiobjective optimization and hybrid evolutionary algorithm to solve constrained optimization problems [J]. IEEE Trans. Syst., Man, Cybern. B, Cybern, 2007, 37(3): 560-575.
[46] Y. Wang, Z. Cai, Y. Zhou, et al.. An adaptive tradeoff model for constrained evolutionary optimization [J]. IEEE Trans. On Evol. Comput., 2008, 12(1): 80-92.
[47] Helio J. C. Barbosa, Afonso C. C. Lemonge. A new adaptive penalty scheme for genetic algorithms [J]. Information Sciences, 2003, 156: 215-251.
[48] R. Storn and K. Price. Differential evolution-A simple and efficient heuristic for global optimization over continuous spaces [J]. Journal of Global Optimization, 1997, 11: 341-359.
[49] J. Sun, Q. Zhang, and E. Tsang. DE/EDA: A new evolutionary algorithm for global optimization [J]. Information Sciences, 2005, 169(3): 249-262.
[50] J. Brest, S. Greiner, et al.. Self-adapting control parameters in differential evolution: A comparative study on numerical benchmark problems [J]. IEEE Trans. on Evol. Comput., 2006, 10(6): 646-657.
[51] N. Noman, H. Iba. Accelerating differential evolution using an adaptive local search [J], IEEE Trans. on Evol. Comput., 2008, 12(1): 107-125.
[52] S. Rahnamayan, H. R. Tizhoosh, and M. M. A. Salama. Opposition-based differential evolution [J]. IEEE Trans. on Evol. Comput., 2008, 12(1): 64-79.
[53] Chang S, H-g C. Optimal approximation of linear systems by a differential evolution algorithm [J]. IEEE Trans. on Syst., Man, Cybern. A, 2001, 31(6): 698-707.
[54] U. K. Chakraborty, S. Das, A. Konar. Differential evolution with local neighborhood [C]. 2006 IEEE Congress on Evolutionary Computation, 2006, 2042-2049.
[55] T. Takahama, S. Sakai. Constrained optimization by theεconstrained differential evolution with gradient-based mutation and feasible elites [C]. 2006 IEEE Congress on Evolutionary Computation, 2006, 1-8.
[56] V. L. Huang, A. K. Qin, P. N. Suganthan. Self-adaptive differential evolution algorithm for constrained real-parameter optimization [C]. 2006 IEEE Congress on Evolutionary Computation, 2006, 17-24.
[57] E. Mezura-Montes, J. Velázquez-Reyes, C. A. C. Coello. Modified differential evolution for constrained optimization [C]. 2006 IEEE Congress on Evolutionary Computation, 2006, 25-32.
[58] J. J. Liang, T. P. Runarsson, E. Mezura-Montes, et al. Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization [R]. Technical Report, 2005. http://www.ntu.edu.sg/home/EPNSugan.
[59] E. M. Montes, C. A. C. Coello, and R. L. Becerra. Engineering optimization using a simple evolutionary algorithm [C]. In Proc. 15th Int. Conf. Tools with Artif. Intell., 2003, 149-156.
[60] H. K. Kim, J. K. Chong, K. Y. Park, and D. A. Lowther. Differential evolution strategy for constrained global optimization and application to practical engineering problems [J]. IEEE Trans. on Magnetics, 2007, 43(4): 1565-1568.
[61] Qingfu Zhang and Yiu-Wing Leung. An orthogonal genetic algorithm for multimedia multicast routing [J]. IEEE Trans. on Evol. Comput., 1999, 3(1): 53-62.
[62] Yiu-wing Leung, and Yuping Wang. An orthogonal genetic algorithm with quantization for global numerical optimization [J]. IEEE Trans. on Evol. Comput., 2001, 5(1): 53-62.
[63] Jinn-Tsong Tsai, Tung-Kuan Liu, and Jyh-Horng Chou. Hybrid Taguchi-genetic algorithm for global numerical optimization [J]. IEEE Trans. on Evol. Comput. 2004, 8(4): 365-377.
[64] Shinn-Ying Ho, Li-Sun Shu, and Jian-Hung Chen. Intelligent Evolutionary Algorithms for Large Parameter Optimization Problems [J]. IEEE Trans. on Evol. Comput., 2004, 8(6): 522-540.
[65] Carlos Cotta, Jose Ma Troya. A Hybrid Genetic Algorithm for the 0-1 Multiple Knapsack Problem [C]. Artificial Neural Nets and Genetic Algorithms 3, Springer-Verlag Wien New York 1998, 250-254.
[66] Jian-cong Bai, Hui-you Chang, Yang Yi. A Partheno-Genetic Algorithm for Multidimensional Knapsack Problem [C]. Proceedings of the Fourth International Conference on Machine Learning and Cybernetics, 2005, 2962-2965.
[67] Ng Chi-Kong, Zhang Lian-Sheng, Li Duan, and Tian Wei-Wen. Discrete filled function method for discrete global optimization [J]. Computational Optimization and Applications, 2005, 31: 87-115.
[68] C. Mohan, H. T. Nguyen. A controlled random search technique incorporating the simulated annealing concept for solving integer and mixed integer global optimization Problems [J]. Computational Optimization and Applications, 1999, 14: 103-132.
[69] K. E. Parsopoulos, and M. N. Vrahatis. Recent approaches to global optimization problems through particle swarm optimization [J]. Natural Computing, 2002, 1: 235-306.
[70] Lino Costa, Pedro Oliveira. Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems [J]. Computers and Chemical Engineering, 2001, 25: 257–266.
[71] Y. C. Lin, and K. S. Hwang. A mixed-coding scheme of evolutionary algorithms to solve mixed-integer nonlinear programming problems [J]. Computers and Mathematics with Applications, 2004, 47: 1295-1307.
[72] I. E. Grossmann. Review of nonlinear mixed-integer and disjunctive programming techniques [J]. Optimization and Engineering, 2002, 3: 227-252.
[73] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu. Finding multiple solutions to general integer linear programs [J]. European Journal of Operational Research, 2008, 184: 802-809.
[74] L. T. Biegler, I. E. Grossmann. Retrospective on optimization [J]. Computers and Chemical Engineering, 2004, 28: 1169–1192.
[75] I. E. Grossmann, L. T. Biegler. Future perspective on optimization [J]. Computers and Chemical Engineering, 2004, 28: 1193–1218.
[76]朱文兴.整数规划的一类填充函数算法[J].应用数学学报, 2000, 23 (4): 481-487.
[77]方开泰,马长兴.正交与均匀试验设计[M].北京:科学出版社, 2001.
[78]邢文训,谢金星.现代优化计算方法[M].北京:清华大学出版社, 2000.
[79]陈志平等.求解中大规模复杂凸二次整数规划问题的新型分枝定界算法[J].计算数学, 2004, 26 (4): 445-458.
[80]杜海峰,刘若辰,焦李成等.求解0-1背包问题的人工免疫抗体修正克隆算法[J].控制理论与应用, 2005, 22 (3): 348-352.
[81]王宇平,焦永昌,张福顺.解多目标优化的均匀正交遗传算法[J].系统工程学报, 2003, 18 (6): 481-486.
[82] Rainer Storn. System design by constraint adaptation and differential evolution [J]. IEEE Trans. on Evolutionary Computation, 1999, 3 (1): 22-34.
[83] Karin Zielinski, and Rainer Laur. Constrained single-objective optimization using differential evolution [C]. 2006 IEEE Congress on Evolutionary Computation, CEC2006, Vancouver, BC, Canada, 2006, pp. 223-230.
[84] M. F. Tasgetiren, and P. N. Suganthan. A multi-populated differential evolution algorithm for solving constrained optimization problem [C]. 2006 IEEE Congress on Evolutionary Computation, CEC2006, Vancouver, BC, Canada, 2006, pp. 33-40.
[85] Janez Brest, Viljem Zumer, and Mirjam Sepesy Mau?ec. Self-adaptive differential evolution algorithm in constrained real-parameter optimization [C].2006 IEEE Congress on Evolutionary Computation, CEC2006, Vancouver, BC, Canada, 2006, pp. 215-222.
[86] G. Rudolph. Convergence analysis of canonical genetic algorithms [J]. IEEE Trans. On Neural Networks, 1994, 5 (1): 96-101.
[87] N. Srinivas, and K. Deb. Multiobjective function optimization using nondominated sorting genetic algorithms [J]. Evolutionary Computation, 1995, 2(3): 221-248.
[88] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan. A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ[J]. IEEE Transactions on Evolutionary Computation, 2002, 6(2): 182-197.
[89] E. Zitzler, L. Thiele. Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach [J]. IEEE Trans. On Evolutionary Computa- tion, 1999, 3(4): 257-271.
[90] J. D. Knowles and D. W. Corne. Approximating the nondominated front using the pareto archived evolution strategy [J]. Evolutionary Computation, 2000, 8 (2): 149-172.
[91] Carlos A. Coello Coello. Evolutionary multi-objective optimization : a historical view of the field [J]. IEEE Computational Intelligence Magazine, 2006, 1 (1): 28-36.
[92] J. F. Bard. Practical bilevel optimization: algorithms and application [M]. The Netherlands: Kluwer Academic Publishers, 1998.
[93] S. R. Hejazi, A. Memariani, G. Jahanshanloo, M. M. Sepehri. Linear bilevel programming solution by genetic algorithm [J]. Computers & Operations Research, 2002, 29: 1913-1925.
[94] G. Anandalingam and D. J. White. A solution for the linear static Stackelberg problem using penalty function [J]. IEEE Transactions Automatic Control, 1990, 35: 1170-1173.
[95] P. Hansen, B. Jaumard and G. Savard. New branch-and-bound rules for linear bilevel programming [J]. SIAM Journal on Science and Statistical Computing, 1992, 13(5): 1194-1217.
[96] Y. H. Liu, and T. H. Spencer. Solving a bilevel linear program when the inner decision maker controls few variables [J]. European Journal of Operational Research, 1995, 81: 644-651.
[97] K. Moshirvaziri, M. A. Amouzegar and S. E. Jacobsen. Test problem construction for linear bilevel programming problems [J]. Journal of Global Optimization 1996, 8: 235-243.
[98] H. Tuy, A. Migdalas, N. T. Hoai-Phuong. A novel approach to bilevel nonlinear programming [J]. Journal of Global Optimization, 2007, 38: 527-554.
[99] Q. Wang, F. Yang, S. Wang and Y. H. Liu. Bilevel programs with multiple followers [J]. Systems Science and Mathematical Sciences, 2000, 13(3): 265-276.
[100] K. M. Lan, U. P. Wen, H. S. Shih and E. S. Lee. A hybrid neural network approach to bilevel programming problems [J]. Applied Mathematics Letters, 2007, 20(8): 880-884.
[101] J. Lu, C. Shi, G. Zhang, Tharam Dillon. Model and extended Kuhn-Tucker approach for bilevel multi-follower decision making in a referential-uncooperative situation [J]. Journal of Global Optimization, 2007, 38: 597-608.
[102] Z. H. Gümüs, and C. A. Floudas. Global optimization of nonlinear bilevel programming problems [J]. Journal of Global Optimization,2001, 20: 1-31.
[103] J. Bard. Convex two-level optimization [J]. Mathematical Programming, 1988, 40: 15-27.
[104] J. Bard and J. Moore. A branch and bound algorithm for the bilevel programming problem [J]. SIAM Journal on Scientific Statistical Computing, 1990, 11: 281-292.
[105] T. Edmunds and J. Bard. Algorithms for nonlinear bilevel mathematical programming [J]. IEEE Trans. on Systems, Man, and Cybernetics, 1991, 21: 83-89.
[106] L. D. Muu, N. V. Quy. A global optimization method for solving convex quadratic bilevel programming problems [J]. Journal of Global Optimization, 2003, 26: 199-219.
[107] K. Shimizu, and E. Aiyoshi. A new computational method for Stackelberg and min-max problems by use of a penalty method [J]. IEEE Transactions on Automatic Control, 1981, 26(2): 460-466.
[108] J. V. Outrata. On the numerical solution of a class of Stackelberg problems [J]. Zeitschrift Fur Operations Research, 1990, 34: 255-277.
[109] V. Oduguwa, and R. Roy. Bi-level optimization using genetic algorithm [C]. In proc. IEEE int. Conf. Artificial Intelligence Systems, 2002, 123-128.
[110] J. E. Falk, Jiming Liu. On bilevel programming, Part I: general nonlinear cases [J]. Mathematical Programming, 1995, 70: 47-72.
[111] P.H. Calamai and L.N. Vicente. Generating quadratic bilevel programming test problems [J]. ACM Transactions on Mathematical Software, 1994, 20(1): 103-119.
[112] L. N. Vicente and P. H. Calamai, Bilevel and multilevel programming: A bibliography review [J]. Journal of Global Optimization, 1994, 5(3): 291-306.
[113] S. Dempe. Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints [J]. Optimization, 2003, 52: 333-359.
[114] B. Colson, P. Marcotte, and G. Savard. Bilevel programming: a survey [J]. A Quarterly Journal of Operations Research(4OR), 2005, 3: 87-107.
[115] B. Colson, P. Marcotte, and G. Savard. An overview of bilevel optimization [J]. Ann. Oper. Res., 2007, 153: 235-256.
[116] BaoDing Liu. Stackelberg-Nash equilibrium for multilevel programming with multiple followers using genetic algorithms [J]. Computers Math. Applic., 1998, 36 (7): 79-89.
[117] Yuping Wang, Yong-Chang Jiao, Hong Li. An evolutionary algorithm for solving nonlinear bilevel programming based on a new constraint-handing scheme [J]. IEEE Trans. on Systems, Man, and Cybernetics-Part C, 2005, 35(2): 221-232.
[118] Beno?t Colson, Patrice Marcotte, and Gilles Savard. A trust-region method for nonlinear bilevel programing: algorithm and computational experience [J]. Computational optimization and applications, 2005, 30: 211-227.
[119] B. Colson. BIPA (Bilevel programming with approximation methods): Software guide and test problems [R]. Technical Report CRT-2002-38, Centre de Recherche sur les Transports, Universitéde Montréal, Montréal, Qc, Canada, Oct. 2002. (http//:citeseer. ist. psu.edu/ Cache).
[120] E. Aiyoshi, K. Shimizu. A solution method for the static constrained Stackelberg problem via penalty method [J]. IEEE Trans. on Automatic Control, 1984, 29(12): 1111-1114.
[121] M. A. Amouzegar. A global optimization method for nonlinear bilevel programming problems [J]. IEEE Trans. on Systems, Man, and Cybernetics-Part B: Cybernetics, 1999, 29(6): 771-777.
[122] H. I. Calvete, C. Gal. The bilevel linear/linear fractional programming problem [J]. European Journal of Operational Research, 1999, 114(1): 188-197.
[123] Z. H. Gümüs, C. A. Floudas. Global optimization of mixed-integer bilevel programming problems [J]. Computational Management Science, 2005, 2: 181-212.
[124] J. T. Moore, J. F. Bard. The mixed integer linear bilevel programming problem [J]. Operations Research, 1990, 38: 911-921.
[125] U. P. Wen, and Y. H. Yang. Algorithms for solving the mixed integer two-level linear programming problem [J]. Computers & Operations Research, 1990, 17: 133-142.
[126] U. P. Wen, A. D. Huang. A simple Tabu Search method to solve the mixed-integer linear bilevel programming problem [J]. European Journal of Operational Research, 1996, 88: 563-571.
[127] L. Vicente, G. Savard and J. Judice. Discrete linear bilevel programming problem [J]. Journal of Optimization theory and Applications, 1996, 89(3): 597-614.
[128] J. F. Bard, and J. T. Moore, An algorithm for the discrete bilevel programming problem [J]. Naval Research Logistics, 1992, 39: 419-435.
[129] S. Dempe, V. Kalashnikov, R. Z. Ríos-Mercado. Discrete bilevel programming: Application to a natural gas cash-out problem [J]. European Journal of Operational Research, 2005, 166: 469-488.
[130] T. A. Edmunds, and J. F. Bard. An algorithm for the mixed-integer nonlinear bilevel programming problem [J]. Annals of Operations Research, 1992, 34: 149-162.
[131] R. H. Jan, and M. S. Chern. Non-linear integer bilevel programming [J]. European Journal of Operational Research, 1994, 72: 574-587.
[132] L. Vicente, G. Savard and J. Judice. Descent approach for quadratic bilevel programming [J]. J. Optim. Theory and Appl., 1994, 81(2): 379-399.
[133] C. Kolstad, and L. Lasdon. Derivative evaluation and computational experience with large bilevel mathematical programs [J]. Journal of Optimization theory and Applications, 1990, 65: 485-499.
[134] E. Aiyoshi, and K. Shimizu. Hierarchical decentralized systems and its newsolution by a barrier method [J]. IEEE Trans. on Systems, Man, and Cybernetics, 1981, 11: 444-449.
[135] Y. Ishizuka and E. Aiyoshi. Double penalty method for bilevel optimization problems [J]. Annals of Operations Research, 1992, 34: 73-88.
[136] K. Shimizu and Min Lu. A global optimization Method for the Stackelberg problem with convex functions via problem transformation and concave programming [J]. IEEE Trans. on Systems, Man, and Cybernetics, 1995, 25(12): 1635-1640.
[137] J. F. Bard. Some properties of the bilevel programming problem [J]. Journal of Optimization Theory and Applications, 1991, 68(2): 371-378.
[138] A. Yezza. First-order necessary optimality conditions for general bilevel programming problems [J]. Journal of Optimization Theory and Applications, 1996, 89(1): 189-219.
[139] Y. Ishizuka. Optimality conditions for quasi-differentiable programs with application to two-level optimization [J]. SIAM J. Control and Optimization, 1988, 26(6): 1388-1398.
[140] H. I. Calvete, and C. Gale. On the quasiconcave bilevel programming problem [J]. Journal of Optimization Theory and Applications, 1998, 98(3): 613-622.
[141] J.V. Outrata. Necessary optimality conditions for Stackelberg problems [J]. Journal of Optimization Theory and Applications, 1993, 76(2): 305-320.
[142] K. Shimizu, and Y. Ishizuka. Optimality conditions and algorithms for parameter design problems with two-level structure [J]. IEEE Trans. On Automatic Control, 1985, 30(10): 986-993.
[143] P. A. Clark, and A. W. Westerberg. Bilevel programming for steady-state chemical process design-I: Fundamentals and Algorithms [J]. Computers Chem. Engng., 1990, 14(1): 87-97.
[144] P. A. Clark. Bilevel programming for steady-state chemical process design-Ⅱ: Performance study for Nondegenerate problems [J]. Computers Chem. Engng., 1990, 14(1): 99-109.
[145] Z. Gao, H. Sun, L. L. Shan. A continuous equilibrium network design model and algorithm for transit systems [J]. Transportation Research Part B, 2004, 38: 235-250.
[146] Suh-Wen Chiou. Bilevel programming for the continuous transport network design problem [J]. Transportation Research Part B, 2005, 39: 361–383.
[147]向丽.递阶优化问题理论及其算法研究与进展[J].控制与决策, 2001,16(6): 854-858.
[148]杨若黎,顾基发.一类非线性两级规划问题的模拟退火求解[J].系统工程理论与实践, 1997, 17(7):52-64.
[149] S. Dempe. A bundle algorithm applied to bilevel programming problems with non-unique lower level solutions [J]. Computational Optimization and Applications, 2000, 15: 145-166.
[150]魏祥云,王先甲.二层系统决策的最优性[J].系统工程理论与实践, 1998, 6: 7-12.
[151]谈烨,仲伟俊,徐南荣.多种资源在多项目间分配的两层决策方法[J].系统工程学报, 1999, 14(3): 290-295.
[152]杜纲.非光滑多目标Stackelberg问题的最优性条件[J].系统工程学报, 1996, 11(2): 22-28.
[153] A. G. Mersha, S. Dempe. Linear bilevel programming with upper level constraints depending on the lower level solution [J]. Applied Mathematics and Computation, 2006, 180: 247-254.
[154] G. S. Liu, J. Y. Han, J. Z. Zhang. Exact penalty functions for convex bilevel programming problems [J]. Journal of Optimization Theory and Applications, 2001, 110 (3): 621-643.
[155]邓先礼,雷万明.用罚函数求解二层凸规划的方法[J].应用数学学报,2001, 24 (2): 161-167.
[156]向丽,顾培亮.一种基于遗传算法的双层非线性多目标决策方法[J].系统工程理论方法应用, 1999, 8(3): 16-21.
[157]王广民,万仲平,王先甲.二(双)层规划综述[J].数学进展, 2007, 36 (5): 513-529.
[158]盛昭翰,吕庆喆,徐南荣.一类二层决策问题的性质分析和基于Frank-Walfe方法的神经网络算法[J].自动化学报, 1996, 22 (6): 657-665.
[159]刘树安,尹新,郑秉霖等.二层线性规划问题的遗传算法求解[J].系统工程学报, 1999, 14 (3): 280-285.
[160]仲伟俊,徐南荣.两层决策的波尔兹曼机方法[J].系统工程学报, 1995, 10(1): 7-13.
[161]王春峰,李光泉,郑丕谔.非光滑两级优化问题的必要条件及其算法[J].系统工程学报, 1998, 13 (3): 92-99.
[162] C. Audet, J. Haddad, G. Savard. A note on the definition of a linear bilevel programming solution [J]. Applied Mathematics and Computation, 2006, 181: 351-355.
[163] C. Shi, J. Lu, G. Zhang, et al. An extended branch and bound algorithm for linear bilevel programming [J]. Applied Mathematics and Computation, 2006, 180: 529-537.
[164] C. Shi, J. Lu, G. Zhang. An extended Kth-best approach for linear bilevel programming [J]. Applied Mathematics and Computation, 2005, 164: 843-855.
[165]林丹,王宏,李敏强.用多目标进化算法求解二层规划双目标模型[J].系统工程理论与实践, 2006, 5: 106-110.
[166]李宏,王宇平.解非线性两层规划的一种混合遗传算法[J].西安电子科技大学学报, 2002, 29(6): 840-843.
[167] Zhi Xu, Hong Li, Qi-zhong Liu and Jian-ying Li. Pattern synthesis of conformal antenna array by the hybrid genetic algorithm [J]. Progress in Electromagnetics Research (PIER), 2008, 79: 75-90.
[168] Shamim Akhtar, Kang Tai and Tapabrata Ray. A socio-behavioural simulation model for engineering design optimization [J]. Eng. Opt., 2002, 34: 341–354.
[169] H. I. Calvete, C. Galé, P. M. Mateo. A new approach for solving linear bilevel problems using genetic algorithms [J]. European Journal of Operational Research, 2008, 188: 14-28.
[170] C. Audet, G. Savard, W. Zghal. New Branch-and-Cut Algorithm for Bilevel Linear Programming [J]. J. Optim. Theory Appl. ,2007, 134: 353-370.
[171]左黎明,汤鹏志,廖宇波等.关于销售集团投资设置销售分店问题IP模型[J].数学的实践与认识, 2005, 35 (2): 21-25.
[172]姚瑞枫,宋玉阶.多维0-1背包问题的混合遗传算法[J].武汉科技大学学报(自然科学版),2003,26(2):214-217.
[173]周水生,刘三阳,刘红英.价格控制问题及其推广形式的罚函数法[J].系统工程学报, 1999, 14(2): 156-161.
[174]阮国桢,杨丰梅,汪寿阳.线性二级价格控制问题的单纯形算法[J].系统工程理论与实践, 1996, 12: 38-43.
[175]滕春贤,姚锋敏,陈东彦.价格控制问题精确罚等价形式的研究[J].黑龙江大学自然科学学报, 2007, 24(1): 10-15.
[176] W. Bialas,M. Karwan. Two-level linear programming [J]. Management Science, 1984, 30 : 1004-1020.
[177]李煜华,张铁男,孙凯.基于价格控制的排污收费二层线性规划模型[J].统计与决策, 2008, 11: 41-43.
[178] M. Labbé, P. Marcotte, and G. Savard. A bilevel model of taxation and its applications to optimal highway pricing [J]. Management Science, 1998, 44: 1595-1607.
[179] P. Marcotte. Network design problem with congestion effects: A case of bilevel programming [J]. Mathematical Programming, 1986, 34:142–162.
[180] A. Migdalas. Bilevel programming in traffic planning: models, methods and challenge [J]. Journal of Global Optimization, 1995, 7: 381-405.
[181] Janet Clegg, Mike Smith, Yanling Xiang, et al. Bilevel programming applied to optimising urban transportation [J]. Transportation Research Part B, 2001, 35: 41-70.
[182] W. Candler, R. Townsley. A linear two-level programming problem [J]. Computers and Operations Research, 1982, 9: 59–76.
[183]刘红英,刘三阳.有关线性价格控制问题的注记[J].系统工程理论与实践, 1999, 19 (4): 141-143.
[184]刘国山,韩继业,汪寿阳.双层优化问题的信赖域算法[J].科学通报, 1998, 43 (4): 383-387.
[185] P. Marcotte, G. Savard, D. L. Zhu. A trust region algorithm for nonlinear bilevel programming [J]. Operations Research Letters, 2001, 29: 171-179.
[186]李云雁等.实验设计与数据处理[M].北京:化学工业出版社, 2005.
[187] P. C. Chu, and J. E. Beasley. A genetic algorithm for the multidimensional knapsack problem [J]. Journal of Heuristics, 1998, 4: 63–86.
[188] D. H. Wolpert, and W. G. Macready. No free lunch theorems for optimization [J]. IEEE Trans. Evol. Comput., 1997, 1(1): 67-82.
[189] K. Deb. An efficient constraint handing method for genetic algorithms [J]. Computer Methods in Applied Mechanics and Engineering, 2000, 186 (2/4): 311-338.
[190]方开泰,王元.数论方法在统计中的应用[M].北京:科学出版社, 1996.
[191]高齐圣,潘德惠.基于均匀设计的遗传算法及其应用[J].信息与控制, 1999, 28(3): 236-240.
[192] S. Dempe. A necessary and a sufficient optimality condition for bilevel programming problems [J]. Optimization, 1992, 25: 341-354.
[193] D. L. Zhu, Q. Xu, Z. Lin. A homotopy method for solving bilevel programming problem [J]. Nonlinear Analysis, 2004, 57: 917-928.
[194] S. Dempe. First-Order Necessary Optimality Conditions for General Bilevel Programming Problems [J]. Journal of Optimization Theory and Applications, 1997, 95(3): 735-739.
[195] N. P. Faísca, V. Dua, B. Rustem, et al. Parametric global optimisation for bilevel programming [J]. Journal of Global Optimization, 2007, 38: 609-623.
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